The story
From the former practice of radio presenters asking listeners to mail in answers to competitions on postcards, these being easier to process in volume since no envelope was involved.
Etymology from Wiktionary, CC BY-SA 4.0.
answers on a postcard
Today it means: Emphasising that the speaker has no idea of the answer to a question, especially an outlandish question (but with no expectation that answers will literally be sent by mail). Full definition →